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Intelligent backpropagated neural networks application on Couette-Poiseuille flow of variable viscosity in a composite porous channel filled with an anisotropic porous layer

Timir Karmakar, Amrita Mandal

TL;DR

This work investigates Couette–Poiseuille flow with depth‑dependent viscosity in a channel partially filled with an anisotropic Brinkman–Forchheimer–Darcy porous layer, coupling Navier–Stokes in the free fluid to the nonlinear Brinkman–Forchheimer equations in the porous medium. The authors derive nondimensional governing equations, obtain large‑ and small‑Darcy‑number asymptotics, and propose an iterative, linearized approximate solution, complemented by an artificial Levenberg–Marquardt backpropagated neural network (ALMM–BNN) to predict intermediate‑$Da$ behavior. They demonstrate that the asymptotic solutions agree with numerical results at the extremes, while the ALMM–BNN captures the correct trends in the intermediate regime and the linearized approach extends applicability, supported by a 3‑layer neural network trained on asymptotic data. The results provide insights into shear‑stress distributions in arterial flows with variable viscosity and anisotropic glycocalyx‑like layers, offering a predictive framework for vascular biomechanics and microfluidic device design.

Abstract

This study examines Couette-Poiseuille flow of variable viscosity within a channel that is partially filled with a porous medium. To enhance its practical relevance, we assume that the porous medium is anisotropic with permeability varying in all directions, making it a positive semidefinite matrix in the momentum equation. We assume the Navier-Stokes equations govern the flow in the free flow region, while the Brinkman-Forchheimer-extended Darcy's equation governs the flow inside the porous medium. The coupled system contains a nonlinear term from the Brinkman-Forchheimer equation. We propose an approximate solution using an iterative method valid for a wide range of porous media parameter values. For both high and low values of the Darcy number, the asymptotic solutions derived from the regular perturbation method and matched asymptotic expansion show good agreement with the numerical results. However, these methods are not effective in the intermediate range. To address this, we employ the artificial Levenberg-Marquardt method with a back-propagated neural network (ALMM-BNN) paradigm to predict the solution in the intermediate range. While it may not provide the exact solutions, it successfully captures the overall trend and demonstrates good qualitative agreement with the numerical results. This highlights the potential of the ALMM-BNN paradigm as a robust predictive tool in challenging parameter ranges where numerical solutions are either difficult to obtain or computationally expensive. The current model provides valuable insights into the shear stress distribution of arterial blood flow, taking into account the variable viscosity of the blood in the presence of inertial effects. It also offers a framework for creating glycocalyx scaffolding and other microfluidic systems that can mimic the biological glycocalyx.

Intelligent backpropagated neural networks application on Couette-Poiseuille flow of variable viscosity in a composite porous channel filled with an anisotropic porous layer

TL;DR

This work investigates Couette–Poiseuille flow with depth‑dependent viscosity in a channel partially filled with an anisotropic Brinkman–Forchheimer–Darcy porous layer, coupling Navier–Stokes in the free fluid to the nonlinear Brinkman–Forchheimer equations in the porous medium. The authors derive nondimensional governing equations, obtain large‑ and small‑Darcy‑number asymptotics, and propose an iterative, linearized approximate solution, complemented by an artificial Levenberg–Marquardt backpropagated neural network (ALMM–BNN) to predict intermediate‑ behavior. They demonstrate that the asymptotic solutions agree with numerical results at the extremes, while the ALMM–BNN captures the correct trends in the intermediate regime and the linearized approach extends applicability, supported by a 3‑layer neural network trained on asymptotic data. The results provide insights into shear‑stress distributions in arterial flows with variable viscosity and anisotropic glycocalyx‑like layers, offering a predictive framework for vascular biomechanics and microfluidic device design.

Abstract

This study examines Couette-Poiseuille flow of variable viscosity within a channel that is partially filled with a porous medium. To enhance its practical relevance, we assume that the porous medium is anisotropic with permeability varying in all directions, making it a positive semidefinite matrix in the momentum equation. We assume the Navier-Stokes equations govern the flow in the free flow region, while the Brinkman-Forchheimer-extended Darcy's equation governs the flow inside the porous medium. The coupled system contains a nonlinear term from the Brinkman-Forchheimer equation. We propose an approximate solution using an iterative method valid for a wide range of porous media parameter values. For both high and low values of the Darcy number, the asymptotic solutions derived from the regular perturbation method and matched asymptotic expansion show good agreement with the numerical results. However, these methods are not effective in the intermediate range. To address this, we employ the artificial Levenberg-Marquardt method with a back-propagated neural network (ALMM-BNN) paradigm to predict the solution in the intermediate range. While it may not provide the exact solutions, it successfully captures the overall trend and demonstrates good qualitative agreement with the numerical results. This highlights the potential of the ALMM-BNN paradigm as a robust predictive tool in challenging parameter ranges where numerical solutions are either difficult to obtain or computationally expensive. The current model provides valuable insights into the shear stress distribution of arterial blood flow, taking into account the variable viscosity of the blood in the presence of inertial effects. It also offers a framework for creating glycocalyx scaffolding and other microfluidic systems that can mimic the biological glycocalyx.

Paper Structure

This paper contains 10 sections, 1 theorem, 47 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Assumptions on the current model and boundary conditions
  4. Non-dimensionalization of governing equations and boundary conditions
  5. Asymptotic solutions
  6. Solution for large Darcy number
  7. Solution for small Darcy number
  8. Proposed approximate solution
  9. Results and discussion
  10. Conclusions

Key Result

Theorem 1

(see p. 439 in Horn2012matrix) Let $A$ be a $n\times n$ Hermitian and positive definite matrix, and let $m\in \{2,3,4...\},$ then there exists a unique Hermitian positive semidefinite matrix $B$ such that $B^{m}=A$.

Figures (17)

  • Figure 1: Schematic diagram of the physical situation
  • Figure 2: (a) $|\Phi'(u_{p})|$ behavior for $0\leq y \leq \lambda$; (b) $|\Omega'(u_{f})|$ behavior for $\lambda\leq y \leq 1$; when $Q=1, M=1, f=1, \beta=0.5, K=1, \phi=0, \lambda=0.5, F=10, Da=0.01,$ and $U=1.$
  • Figure 3: Dimensionless velocity profile (a) for $\lambda=0.5$, $Q=1$, $M=1$, $\beta=0$, $Da=0.001$, $K=1$, $F=1$, $U=1$, $f=1$; (b) for $\lambda=0.5$, $Q=1$, $M=1$, $\beta=0.5$, $Da=1$, $K=1$, $F=1$, $U=1$, $f=1$.
  • Figure 4: Dimensionless velocity profile (a) for different $K$ when $\lambda=0.5$, $Q=1$, $M=1$, $\beta=0.5$, $Da=0.01$, $F=1$, $U=1$, $f=1$, and $\phi=0$; (b) for different $\phi$ when $\lambda=0.5$, $Q=1$, $M=1$, $\beta=0.5$, $Da=0.01$, $K=0.5$, $F=1$, $U=1$, and $f=1$.
  • Figure 5: Dimensionless velocity profile (a) for different $Da$ when $\lambda=0.5$, $Q=1$, $M=1$, $\beta=0.5$, $K=1$, $F=1$, $U=1$, $f=1$, $\phi=0$; (b) for different $F$ when $\lambda=0.5$, $Q=1$, $M=1$, $\beta=0.5$, $Da=0.1$, $K=1$, $U=1$, $f=1$.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Theorem 1